Autonomous dynamical system approach for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> gravity

In this work we shall investigate the cosmological dynamical system of $f(R)$ gravity, by constructing it in such a way so that it is rendered autonomous. We shall study the vacuum $f(R)$ gravity case, but also the case that matter and radiation perfect fluids are present along with the $f(R)$ gravity. The dynamical system is constructed in such a way that the time dependence of the system is contained in a single parameter which depends on the Hubble rate and its second derivative. The autonomous structure of the dynamical system is achieved when this parameter is constant; therefore we focus on these cases. For the vacuum $f(R)$ case, we investigate two cases with the first leading to a stable de Sitter attractor fixed point but also to an unstable de Sitter fixed point, and the second is related to a matter dominated era. The stable de Sitter attractor is also found for the $f(R)$ gravity in the presence of matter and radiation perfect fluids. In all the cases we perform a detailed numerical analysis of the dynamical system, and we also investigate in detail the stability of the resulting fixed points. Also, we present an exceptional feature of the ${R}^{2}$ gravity model, in the absence of perfect fluids. Finally, we investigate what is the approximate form of the $f(R)$ gravities near the stable and the unstable de Sitter fixed points.

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